Precise Definition of a Limit and Proving - Linear Function Example
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- Опубликовано: 15 сен 2024
- In this example we look at proving a limit result by using the precise definition of a limit.
This definition often confuses people but really we are just trying to capture the notion of 'closeness' by using absolute value and inequalities.
This type of limit example where we deal with a linear function is largely mechanical, so even if you absolutely have no clue what the heck is really going on, you should be able to memorize the steps easily enough!
Note that the notion of a limit of crucially important in calculus.
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yes, i am still doing that and it is really helpful and free
lmao that ad pissed me off so bad.
Oh and Use adblocker.
Don't use adblocker... it demonetizes his videos. If at possible watch the ads and let them play through and show him some support. Beside I don't mind them spending money on a product I'll never buy to support ones I would pay for.
Exactly
7 months later and I get the same ad
Just got it now..
Thank you so much, you are a godsent, my Calculus 1 Professor is making me learn this on my own and you just saved my life
relateable
Fr in that situation rn 😭
I feel you
DUDE THANK YOU SO MUCH THIS IS ABSOLUTELY CRAZY THIS MAKES SO MUCH SENSE NOW.
Glad it helped!
Thankyou so much, my professor didn't do a great job of showing this and now it all makes sense!
I don't know what the difference between my textbook and my professor and you is, but something just clicked watching this. Thank you so much, you are saving my grades.
I feel so bad because I finished all three CALs and now I am in differential equations, but I was never able to master this proof. I just hate the stupid epsilons man.HAHAHAHA. Thank you for the video Patrick.
I really don't know what I would do without you sir. Thank you ver much...
Hey. I now understand how this relates to the overall picture of limits definition that is for every epsilon we have a delta, but things would be even better if we approached this problem from a geometric perspective: getting delta from a graph. And thanks for your videos!
lol it took me a month to figure out derivates means rate of change at point.
@@blacktape52black72 lol XD
so clear! never understand the concept of precise limit but u just cleared that out
I regrettably feel that after finishing and currently studying differential equations, I was never able to fully understand this proof. Thank you to my advisor for pointing me in the right direction.
thank you so much. I was thinking I'm never gonna get this concept ..but after watching this video, I feel like I got it.
This video explains the precise definition better than the other one you showed. thx ✌️
Thank you so much! Wayyyy better explanation than my professor.
Wow, perfect timing to post this...my first calc1 test in a few hours and this proof is the hardest concept of the first section.
Yo how did the test go?
@@uyujnnqaal lol I ended up dropping the class, didn't need this level of calc for my major it turned out
Thanks for all the videos bro. Listen, can you do an example where the inequality cannot be solved algebraically, so you have to "enlarge" the function in order to get something that CAN be solved, then from there prove that the enlarged function is less than epsilon, which proves that the original function we started with is ALSO less than epsilon since its smaller than the function we enlarged. I hope I was able to explain what I'm asking.
This video saved me from poor instruction. Thank you sm
Thank you very much sir!! Your channel helps and motivates mi in school 😊
happy to help you :)
It seems [to me] that this epsilon-delta strategy can only affirm a limit, if the function behaviour is well behaved around the point of discontinuity.
Which presupposes prior knowledge of the behaviour of the function. That would make the proof, self serving, in that it aims at proving a property thats already evident.
By well behaved, I mean that the function is either sequentially increasing/decreasing on either side of the proposed limit.
What about irregular functions that make jumps in F(x) as we get closer to the proposed limits?
I really don't know, so I need some coaching.
May be it would be helpful if we began with a definition of the types of functions where this approach can work or not. TYSM for the vid
Thank you very much Mr Patrick. Now am confident enough that i will muster calculus!!!
Thanks dude, got exam tomorrow and it really helped
I can't express my gratitude 💕💕💕💕
oh my god i love u😭i was struggling so hard
If only I had this last week or my calc1 test!
Saying ' just reverse the steps' is very confusing--you should focus on the implication issue. great video!
Indeed! This is why I couldn't get it. I was like, why in the hell are we doing the same thing, but in reverse order?! of course we're going to get the same thing!
Super explanation sir I could really understand your lecture
i subbed and it was worth it. Thanks Mate!
i don't understand why E/2 = DELTA.
for example:
6
@@bilaltariq7819 you neglect to mention that epsilon is totally arbitrary and delta DEPENDS on epsilon....and how crucially important that is....but still FACTS.
Lol for example if you have a function f(x) = 2x
2x = 10 < 11
and 2x = 4 < 11
SURELY, you ain't gonna say 10 = 4 lol....
It's more like f(epsilon) = delta so to speak.
So since f(epsilon) = delta
we have f(epsilon) = (1/2)E = E/2 = delta.
If he didn't work the other way, it would be a prank. lol A complete s c a m.
Great video thanks man!
YEAHH NOOWWW I UNDERESTAND IT THANK YOU LETS GOOOOOOOOOOO
Shouldn't it be 0 < |x-a| < δ, not just |x-a| < δ, to ensure x≠a?
Yes, that's true. He probably didn't write it to simplify things.
NOW I UNDERSTAND THANK YOU!
Thank you buddy❤
Don't get confused by all the technical terms guys. The simple intuition behind limit is the value approached by a function as the input approaches a certain value.
That's all, the epsilon and delta definition is stating the same thing.
As long as you understand the meaning of what a limit is, you don't need to know the technical confusing terminology.
THIS IS GEM! THANKSSSS!
Thanks! I love your video!
1st : absolute of function is less than e and prove absolute of points less than a manipulation ( related to e which is dell )
2nd : now take that absolute of points less than dell (but put manipulated value) will always implies absolute of function is less than e
But HOW did you decide that delta=epsilon/2?
Sangat Membantu, Terima Kasih ( Indonesia) = So helpfull, Thank You
this video helps me to pull an all-nighter cause my (smart) lecturer :D
amazing .....but did you choose delta to = epsiolon/2 becaue you knew it would work or would it still have worked if say you chose delta to = epsilon/3
It will work with ε/3
suppose delta = ε/3
|x-4| < ε/3
2|x-4| < 2ε/3
|2x-8| < 2ε/3
but 2ε/3 is less then ε (2ε/3 < ε) so..
|2x-8| < ε
see he choose it as ε/2 just to make it easy.
If it works for anything why specifically epsilon/2?
@@sudheerthunga2155 Because epsilon/2 kind of encapsulates all the other cases. Taking from the example above, if you directly deduce that |2x-8|
@@cadu7698 Epsilon /2 is the minimum distance,right??
@@sudheerthunga2155 That would imply that x can be as far as you want it to be from 4, as long as it's E/2 "units" from it, which is a wrong assumption.
Patric u are amazing i've learnt things from u in two nights watching u is also enjoyfull.u deserve fucking oscar prize
When I look through my textbook, they randomly get values for delta and then I get confused because idk how they did it
and now you know?
I sure love that i'm taking calculus as a biology student.
But, this was actually very helpful.
You should be taking Calculus if you are a Biology student though...
Thank you a lot lot lot lot lot lotttttt
Tqsm sir 😍
Thank you
this is beautiful
Im awestruck
thank uh so much sir🥰
Most welcome
@@patrickjmt can you plz make more vedios on this topic🤗. Love from india
helps a lot!!
thanks!
it would help if you could show a case where the limit DOES NOT exist - eg at a step discontinuity.
My professor did it by getting a inequality for f(x) - L from the delta and then he chosed a ephsilent for that delta is it correct? Sorry for my english
makes sense that a left handed person is tutoring calculus XD
Nice explanation.
Hello can you reverse
|x-a|
I don't get why this isn't a tautology. You basically did some math, then you did the opposite steps to get back to where you started and somehow that is a proof?
Here what is meant by Delta and epsilon exactly .can we take any other letter instead of them as which means small +ve number
the only part im confused on is why youre allowed to say L = E / 2 ?
Did you use Differential and Integral Calculus by Love & Rainville as a reference for this video?
Thank sir
very nice
Thank You! :)
In what part of the limit definition it´s said that the point can´t be L? or a ?
Hey Patrick! Just curious, is this your main job, or do you have a day job?
Thx Man :)
Hi,What are u study?
I am confused about the part where we let delta = epsilon/2. Why does this work because wouldn't substituting ABS x-4 make it so that delta is less than epsilon/2
becase abs x-4 = delta.
plz can you show how to prove lim 1/(1-x)^2 = infinity x tends to 1
Why epsilon over 3 over 4 and all will work....???please help
thx
thx bro, IM FROM TUCKER.
One question, how can you be so sure 2 | x - 4 | is the same as | 2 (x -4) | ?
because it could be | (-2) . (x - 4) | as far as i know about absolute number
@Taurus Capricorn 😅😅too bad
Give a smart guy a sharpie and a camera, and he will do wonders!
Cool thank you
Thank yooouuu!
What if you get a problem where a is a negative number?
Watch from 3:40
The philosophizing is necessary
im confused, how is it mathematically proven that "delta" is equal to "epsilon dived by 2?" thanks guys
I know its been awhile since you posted this but the reason delta = epsilon divided by 2 is simply that we are letting delta = epsilon divided by 2. We wanted to find out what we needed to set our delta value equal to so that we could get | 2x + 3 - 11|. If you ever take Calc 3 this will still be useful to know when taking limits in multiple variables.
It just makes it easier to solve
thank you for the explanation. Please use a normal pen or pencil while writing on a paper
what do you mean prove it just does
nice but can you please make a video to help me do better in algebra one thank you
Praise
can't even watch this with his hand in the way of everything written had to pause like 125 times ... leftys
What I learned:
if 1+1=2, then subtract 1 from both sides:
1=2-1
1=1, then let 1 equal to 1,
1=1
add one to both sides:
1+1=1+1
which is also equal to 1+1=2
BOOM here you go, enjoy your accomplishments, you've proved absolutely nothing.
Are you still there?
Well just wanted to say that you are the only comment that made me laugh in this horror movie
Thx for it
AMAZING
😄😄😄😄😄😄😄
god bless
what about the pre-proof?
i want to get solution of the end
bro can you tell me express this definition of limit using quantifiers
can epsilon be negative? i ended up with -E/3 on one problem.
epsilon is defined to be a small positive number
i found that out the hardway lol, thanks!
This epic
a?
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Güzel anlatım 👍
Bizim yarak kafalı hocalardan bin kat iyi
@@vahsibatljohnny6682 😂
I understood everything except for the part where you said we could use epsilon/3 or epsilon/4 and it would still work. At that one moment you completely lost me
To prove that this limit exists and satisfy the definition, we need to establish a relationship between δ and ε such that, if x is within δ of a, f(x) is within ε of L. For example, say we find that δ is ε/2 for a particular limit, and say we arbitrarily pick 4 for ε. This means that, in order for f(x) to be no more than 4 away from L, x must be no more than 2 from a. If we change δ to ε/4, this definition is still satisfied because for an ε of 4, we have a δ of 1, and an x no more than 1 away from a still produces an f(x) no more than 4 from L. We can make δ whatever we want as long as it is not more than ε/2. Basically, we can make the tolerances for δ as strict as we like, but we cannot make them more lenient. If δ is greater than ε/2 in this example, f(x) will no longer be within the desired distance of L, and the definition cannot be satisfied.
delta = 0.00000000000000001 and epsilon = 0.000000000000000002 make a tiny difference.
quentin tarantino?
So hard to see anything with your hand in the way. I had to keep rewinding to see what was under your hand. What should have been around a 7 minute video took me nearly 30 minutes to watch. Also, there's only one example and more are needed to fully understand this topic.
You are reading from thomas calculas
😸
god this is never going to make any sense to me. i give up on precise definition of a limit. this is just totally beyond my INTELLIGENCE.
Well, I kind of started getting it just now, and I've attended around four Calculus courses... so keep trying and someday it will make sense (that's not supposed to sound pessimistic).
wtf